The base code \(\mathcal {C}_0\) . Set \(\varepsilon _0 = 1/d_2^2 = \lambda _2^4/b_2^4 \le \lambda _2^4/3\)  (this
choice differs from Ta-Shma's and it appears because we are
essentially working with \(H^2\)  rather than \(H\) ). We will choose a base code \(\mathcal {C}_0\)  such that the desired code will be obtained as a direct sum lifting of \(\mathcal {C}_0\) , and because this lifting preserves the dimension, the dimension of \(\mathcal {C}_0\)  should be \(D\) . We choose \(\mathcal {C}_0\)  to be an
\(\varepsilon _0\) -balanced code with dimension \(D\)  and block length \(n=O_{\varepsilon _0}(D)\) .
For instance, we can start with
any good (constant rate and relative distance) linear base code
\(\mathcal {C}_0\)  that has an efficient unique decoding algorithm and obtain a
\(\varepsilon _0\) -balanced lifted code that can be efficiently unique decoded (as
long as \(\varepsilon _0\)  is constant) using the framework in [1]}.
